Put the potato in a bowl, cover with cling film and microwave on high for 4 mins until partially softened, then leave to cool slightly. Meanwhile, grate the apple and sprouts, to their cores, into a clean tea towel, then grate the potato on top and use the tea towel to squeeze out as much moisture as possible. Tip into a bowl, crumble in the black pudding, add the garlic and stir through the cumin seeds. Season generously.
Heat 1½ tbsp oil in a frying pan over a medium heat. Add half the potato mix, flatten with the back of a spatula to a thickness of around 1½ cm and fry for 3 mins. Put a plate on top of the pan, flip the rösti over, slide back into the pan and cook for 3 mins on the other side. Once golden, set aside, covered with some kitchen foil, and repeat with the remaining mixture to make a second rösti.
Meanwhile, use a separate pan to fry the two eggs in a little oil. Serve each rösti topped with a fried egg, some cracked black pepper and a scattering of parsley.